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MIMO Detection Methods: How They Work
Erik G. Larsson
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Erik G. Larsson, MIMO Detection Methods: How They Work, 2009, IEEE signal processing
magazine, (26), 3, 91-95.
http://dx.doi.org/10.1109/MSP.2009.932126
Postprint available at: Linköping University Electronic Press
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[
lecture
NOTES
]
Digital Object Identifier 10.1109/MSP.2009.932126
I
n communications, the receiver
often observes a linear superposition
of separately transmitted informa-
tion symbols. From the receiver’s
perspective, the problem is then to
separate the transmitted symbols. This is
basically an inverse problem with a
finite-alphabet constraint. This lecture
will present an accessible overview of
state-of-the-art solutions to this problem.
RELEVANCE
The most important motivating applica-
tion for the discussion here is receivers
for multiple-antenna systems such as
multiple-input, multiple-output (MIMO),
where several transmit antennas simul-
taneously send different data streams.
However, essentially the same problem
occurs in systems where the channel
itself introduces time- or frequency-dis-
persion, in multiuser detection, and in
cancellation of crosstalk.
PREREQUISITES
General mathematical maturity is
required along with knowledge of basic
linear algebra and probability.
PROBLEM STATEMENT
Concisely, the problem is to recover the
vector
s
[ R
n
from an observation of
the form
y
5 Hs 1 e,
y
[ R
m
, (1)
where H [
R
m3n
is a known (typically,
estimated beforehand) channel matrix
and e [
R
m
represents noise. We
assume that e , N
1
0, sI
2
. The ele-
ments of
s
, say s
k
, belong to a finite
alphabet
S
of size
|
S
|
. Hence there are
|
S
|
n
possible vectors
s
. For simplicity of
our discussion, we assume that all
quantities are real-valued. This is most-
ly a matter of notation, since
C
n
is iso-
morphic to
R
2
n
. We also assume that
m
$
n, that is, (1) is not underdeter-
mined, and that H has full column
rank. This is so with probability one in
most applications. We also assume that
H has no specific structure. If H has
structure, for example, if it is a Toeplitz
matrix, then one should use algorithms
that can exploit this structure.
We want to detect
s
in the maxi mum-
likelihood (ML) sense. This is equivalent to
The problem:
min
s
[ S
n
7y 2 Hs7. (2)
Problem (2) is a finite-alphabet-con-
strained least-squares (LS) problem,
which is known to be nondeterministic
polynomial-time (NP)-hard. The compli-
cating factor is of course the constraint
s
[
S
n
, otherwise (2) would be just clas-
sical LS regression.
SOLUTIONS
As a preparation, we introduce the
QL-decomposition of H : H 5 QL, where
Q [
R
m3n
is orthonormal (Q
T
Q 5 I),
and
L
[ R
n3n
is lower triangular. Then
7y 2 Hs7
2
5 7QQ
T
1
y2Hs
2
7
2
1 7
1
I2QQ
T
21
y 2Hs
2
7
2
57Q
T
y 2 Ls7
2
1 7
1
I 2 QQ
T
2
y7
2
,
where the last term does not depend on
s
.
It follows that we can reformulate (2) as
Equivalent problem: min
s
[ S
n
7y
|
2 Ls 7,
where
y
|
! Q
T
y
(3)
or, in yet another equivalent form, as
min
5
s
1
,c, s
n
6
s
k
PS
5
f
1
1
s
1
2
1 f
2
1
s
1
, s
2
2
1
c
1 f
n
1
s
1
, c, s
n
26
,
where
f
k
1
s
1
, c, s
k
2
!
a
y
|
k
2
a
k
l
51
L
k, l
s
l
b
2
. (4)
Problem (4) can be visualized as a
decision tree with n 1 1 layers,
|
S
|
branches emanating from each nonleaf
node, and
|
S
|
n
leaf nodes. See Figure 1.
To any branch, we associate a hypotheti-
cal decision on s
k
, and the branch metric
f
k
1
s
1
, c, s
k
2
. Also, to any node (except
the root), we associate the cumulative
metric f
1
1
s
1
2
1
c
1 f
k
1
s
1
, c, s
k
2
,
which is just the sum of all branch met-
rics accumulated when traveling to that
node from the root. Finally, to each node,
we associate the symbols
5
s
1
, c, s
k
6
it
takes to reach there from the root.
Clearly, a naive but valid way of solving
(4) would be to traverse the entire tree to
find the leaf node with the smallest cumu-
lative metric. However, such a brute-force
search is extremely inefficient, since there
are
|
S
|
n
leaf nodes to examine. We will
now review some efficient, popular, but
approximate solutions to (4).
ZERO-FORCING (ZF) DETECTOR
The ZF detector first solves (2), neglect-
ing the constraint
s
[
S
n
s
|
! arg min
s
[ R
n
7y 2 Hs7
5 arg min
s
[ R
n
7y
,
2 Ls 7 5 L
21
y
,
. (5)
Of course,
L
21
does not need to be explic-
itly computed. For example, one can do
Gaussian elimination: take s
|
1
5 y
|
1
/L
1,1
,
then s
|
2
5
1
y
|
2
2 s
|
1
L
2,1
2
/L
2,2
, and so forth.
ZF then approximates (2) by projecting
each s
|
k
onto the constellation
S
Erik G. Larsson
MIMO Detection Methods: How They Work
1053-5888/09/$25.00©2009IEEE
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[
lecture
NOTES
]
continued
s
^
k
5
3
s
|
k
4
! arg min
s
k
[ S
|
s
k
2 s
|
k
|
. (6)
We see that s
|
5 s 1 L
21
Q
T
e, so s
|
in (5) is
free of intersymbol interference. This is
how ZF got its name. However,
unfortunately ZF works poorly unless H is
well conditioned. The reason is that the
correlation between the noises in s
|
k
is
neglected in the projection operation (6).
This correlation can be very strong, espe-
cially if H is ill conditioned (the covariance
matrix of s
|
is S ! s
#
1
L
T
L
2
21
2
. There are
some variants of the ZF approach. For
example, instead of computing s
|
as in (5),
one can use the MMSE estimate (take
s
|
5 E
3
s|
y
4
). This can improve perfor-
mance somewhat, but it does not overcome
the fundamental problem of the approach.
ZF DETECTOR WITH
DECISION FEEDBACK (ZF-DF)
Consider again ZF, and suppose we use
Gaussian elimination to compute s
|
in
(5). ZF-DF [1] does exactly this, with the
modification that it projects the symbols
onto the constellation
S
in each step of
the Gaussian elimination, rather than
afterwards. More precisely,
1) Detect s
1
via s
^
1
5 arg min
s
1
[ S
f
1
1
s
1
2
5
c
y
|
1
L
1,1
d
.
2) Consider s
1
known (s
1
5 s
^
1
) and
detect s
2
via s
^
2
5 arg min
s
2
[ S
f
2
1
s
^
1
, s
2
2
5
c
y
|
2
2 s
^
1
L
2,1
L
2,2
d
.
3) Continue for k 5 3, . . . , n
s
^
k
5 arg min
s
k
[ S
f
k
1
s
^
1
, c, s
^
k21
, s
k
2
5
c
y
|
k
2S
k
21
l5 1
L
k, l
s
^
l
L
k, k
d
.
In the decision-tree perspective,
ZF-DF can be understood as just examin-
ing one single path down from the root.
When deciding on s
k
, it considers
s
1
, c, s
k
21
known and takes the s
k
that
corresponds to the smallest branch met-
ric. Clearly, after n steps we end up at one
of the leaf nodes, but not necessarily in the
one with the smallest cumulative metric.
In Figure 2(a), ZF-DF first chooses
the left branch originating from the root
(since 1 , 5), then the right branch
(since 2 . 1) and at last the left branch
(because 3 , 4), reaching the leaf node
with cumulative metric 1 1 1 1
3
5
5
.
The problem with ZF-DF is error
propagation. If, due to noise, an incor-
rect symbol decision is taken in any of
the n steps, then this error will propa-
gate and many of the subsequent deci-
sions are likely to be wrong as well. In
its simplest form (as explained above),
ZF-DF detects s
k
in the natural order,
but this is not optimal. The detection
order can be optimized to minimize the
effects of error propagation. Not sur-
prisingly, it is best to start with the sym-
bol for which ZF produces the most
reliable result: that is, the symbol s
k
for
which
S
k, k
is the smallest, and then
proceed to less and less reliable sym-
bols. However, even with the optimal
ordering, error propagation severely
limits the performance.
SPHERE DECODING (SD)
The SD [2], [9] first selects a user
parameter R, called the sphere radius. It
then traverses the entire tree (from left
to right, say). However, once it encoun-
ters a node with cumulative metric
larger than R, then it does not follow
down any branch from this node. Hence,
in effect, SD enumerates all leaf nodes
which lie inside the sphere
7y
|
2 Ls 7
2
# R. This also explains the
algorithm’s name.
In Figure 2(b), we set the sphere radi-
us to R 5 6. The SD algorithm then tra-
verses the tree from left to right. When it
encounters the node “7” in the right sub-
tree, for which 7 . 6 5 R, SD does not
follow any branches emanating from it.
Similarly, since 8 . 6, SD does not visit
Root Node
s
1
= −1
s
2
= −1
s
3
= −1 s
3
= −1 s
3
= −1s
3
= +1 s
3
= +1 s
3
= −1s
3
= +1 s
3
= +1
s
2
= −1
s
2
= +1
s
2
= +1
f
1
(−1) = 1
f
2
(−1, −1) = 2
f
3
(. . .) = 4 f
3
(. . .) = 1
343119
Leaves
{1, 1, 1}
{1, 1, −1}{1, −1, 1}{1, −1, −1}{−1, 1, 1}{−1, 1, −1}{−1, −1, 1}{−1, −1, −1}
f
2
(−1, 1) = 1 f
2
(1, −1) = 2 f
2
(1, 1) = 3
f
1
(1) = 5
s
1
= +1
15
72
756108917
3 8
4
[FIG1] Problem (4) as a decision tree, exemplified for binary modulation (S = {–1, +1}, |S| = 2) and n = 3. The branch metrics f
k
(s
1
, . . ., s
k
)
are in blue written next to each branch. The cumulative metrics f
1
(s
1
)+ . . . + f
k
(s
1
, . . . , s
k
) are written in red in the circles representing
each node. The double circle represents the optimal (ML) decision.
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IEEE SIGNAL PROCESSING MAGAZINE [93] MAY 2009
any branches below the node “8” in the
rightmost subtree.
SD in this basic form can be much
improved by a mechanism called prun-
ing. The idea is this: Every time we reach
a leaf node with cumulative metric M, we
know that the solution to (4) must be
contained in the sphere 7y
|
2 Ls 7
2
# M.
So if M , R, we can set R J M, and con-
tinue the algorithm with a smaller sphere
radius. Effectively, we will adaptively
prune the decision tree, and visit much
fewer nodes than those in the original
sphere. Figure 2(c) exemplifies the prun-
ing. Here the radius is initialized to
R 5`, and then updated any time a leaf
node is visited. For instance, when visit-
ing the leaf node “4,” R will be set to
R 5 4. This means that the algorithm
will not follow branches from nodes that
have a branch metric larger than four. In
particular, the algorithm does not exam-
ine any branches stemming from the
node “5” in the right subtree.
The SD algorithm can be improved in
many other ways, too. The symbols can be
sorted in an arbitrary order, and this order
can be optimized. Also, when traveling
down along the branches from a given
node, one may enumerate the branches
either in the natural order or in a zigzag
fashion (e.g., s
k
5
5
25, 23, 21, 21, 3, 5
6
versus s
k
5
5
21, 1, 23, 3, 25, 5
6
). The
SD algorithm is easy to implement,
although the procedure cannot be directly
parallelized. Given large enough initial radi-
us R, SD will solve (2). However, depending
on H, the time the algorithm takes to finish
will fluctuate, and may occasionally be
very long.
FIXED-COMPLEXITY
SPHERE DECODER (FCSD)
FCSD [3] is, strictly speaking, not really
sphere decoding, but rather a clever
combination of brute-force enumeration
and a low-complexity, approximate detec-
tor. In view of the decision tree, FCSD
visits all
|
S
|
r
nodes on layer
r
, where
r
,
0
# r # n is a user parameter. For each
node on layer
r
, the algorithm considers
5
s
1
, ..., s
r
6
fixed and formulates and
solves the subproblem
min
5
s
r11
, c, s
n
6
s
k
[
S
5
f
r11
1
s
1
, c, s
r11
2
1
. . .
1 f
n
1
s
1
, c, s
n
26
. (7)
1
5
21 2
3
4
1343119
1
5
3
2
78
745610
8
9
17
ZF-DF
1
5
21 23
4
1343119
1
5
3
2
78
745610
8
9
17
SD, No Pruning
(here: R = 6)
15
21
23
41 34 31 19
1
5
327
8
7456
10
8
9
17
SD, Pruning
(here: R = ∞)
15
2
1
23
41 3
4
31 19
1
5
327
8
7456
10
8
9
17
FCSD
(here: r = 1)
(a)
(b)
(c)
(d)
[FIG2] Illustration of detection algorithms as a tree search. Solid-line nodes and branches are visited. Dashed nodes and branches are
not visited. The double circles represent the ultimate decisions on
s
. (a) ZF-DF: At each node, the symbol decision is based on choosing
the branch with the smallest branch metric. (b) SD, no pruning: Only nodes with S
n
k
51
f
k
1
s
1
, . . . , s
k
2
#
R are visited. (c) SD, pruning:
Like SD, but after encountering a leaf node with cumulative metric M, the algorithm will set R :
5
M. (d) FCSD: Visits all nodes on the
r th layer, and proceeds with ZF-DF from these.
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[
lecture
NOTES
]
continued
In effect, by doing so, FCSD will reach
down to
|
S
|
r
of the
|
S
|
n
leaves. To form its
symbol decisions, FCSD selects the leaf,
among the leaves it has visited, which
has the smallest cumulative metric
f
1
1
s
1
2
1
c
1 f
n
1
s
1
, c, s
n
2
.
The subproblem (7) must be solved
once for each combination
5
s
1
, ..., s
r
6
,
that is
|
S
|
r
times. FCSD does this approx-
imately, using a low-complexity method
(ZF or ZF-DF are good choices). This
works well because (7) is overdetermined:
there are n observations ( y
|
1
, c, y
|
n
),
but only n 2 r unknowns (s
r11
, c, s
n
).
More precisely, the equivalent channel
matrix when solving (7) will be a tall sub-
matrix of H, which is generally much
better conditioned than H.
Figure 2(d) illustrates the algorithm.
Here
r
5 1. Thus, both nodes “1” and “5”
in the layer closest to the root node are
visited. Starting from each of these two
nodes, a ZF-DF search is performed.
Naturally, the symbol ordering can be
optimized. The optimal ordering is the
one which renders the problem (7) most
well-conditioned. This is achieved by
sorting the symbols so that the most
“difficult” symbols end up near the tree
root. Note that “difficult symbol” is non-
trivial to define precisely here, but intui-
tively think of it as a symbol s
k
for which
S
k,k
is large.
The choice of
r
offers a tradeoff
between complexity and performance.
FCSD solves (2) with high probability
even for small
r
, it runs in constant time,
and it has a natural parallel structure.
Relatives of FCSD that produce soft out-
put also exist [4].
SEMIDEFINITE-RELAXATION
(SDR) DETECTOR
The idea behind SDR [5], [6] is to relax the
finite-alphabet constraint on
s
into a
matrix inequality and then use semidefi-
nite programming to solve the resulting
problem. We explain how it works, for
binary phase-shift keying (BPSK) symbols
(s
k
[
5
6 1
6
). Define
s
2
!
c
s
1
d
, S ! s
2
s
2
T
5
c
s
1
d
3
s
T
1
4
,
C !
c
L
T
L 2 L
T
y
,
2 y
, T
L 0
d
.
Then
7y
|
2 Ls 7
2
5 s
T
cs 1 7y
|
7
2
5 Trace
5
CS
6
1 7y
|
7
2
so solving (3) is the same as finding the
vector s [
S
n
that minimizes Trace {CS}.
SDR exploits that the constraint
s [
S
n
is equivalent to requiring that
rank {S} 5 1, s
n
11
5 1 and diag {S} 5
{1, . . . , 1}. It then proceeds by minimizing
Trace {CS} with respect to
S
, but relaxes
the rank constraint and instead requires
that
S
be positive semidefinite. This
relaxed problem is convex, and can be effi-
ciently solved using so-called interior
point methods. Once the matrix
S
is
found, there are a variety of ways to deter-
mine
s
, for example to take the dominant
eigenvector of
S
(forcing the last element
to unity) and then project each element
onto
S
like in (6). The error incurred by
the relaxation is generally small.
LATTICE REDUCTION (LR)
AIDED DETECTION
The idea behind LR [8], [9] is to trans-
form the problem into a domain where
the effective channel matrix is better
conditioned than the original one. How
does it work? If the constellation
S
is
uniform, then
S
may be extended to a
scaled enumeration of all integers, and
S
n
may be extended to a lattice
S
n
. For
illustration, if S 5
5
23, 2 1, 1, 3
6
, then
S
n
5
5
c, 23, 21, 1, 3, c
6
3
c
3
5
c, 23, 21, 1, 3, c
6
. LR decides
first on an n 3 n matrix
T
that has inte-
ger elements (
T
k,l
[ Z ) and which maps
the lattice
S
n
onto itself:
Ts [
S
n
4s [
S
n
. That is,
T
should be
invertible, and its inverse should have
integer elements. This happens precisely
if its determinant has unit magnitude:
|
T
|
561. Naturally, there are many
such matrices (
T
56I is one trivial
example). LR chooses such a matrix
T
for which, additionally, H
T
is well condi-
tioned. It then computes
s
^
r
! arg min
s9P
S
n
|| y 2
1
HT
2
s
r
||. (8)
Problem (8) is comparatively easy, since
H
T
is well conditioned, and simple
methods like ZF or ZF-DF generally
work well. Once s
^
r
is found, it is trans-
formed back to the original coordinate
system by taking s
^
5 T
21
s
^
r
.
LR contains two critical steps. First, a
suitable matrix
T
must be found. There
are good methods for this (e.g., see refer-
ences in [8], [9]). This is computationally
expensive, but if the channel H stays
constant for a long time then the cost of
finding
T
may be shared between many
instances of (2) and complexity is less of
an issue. The other problem is that while
the solution s
^
always belongs to
S
2
n
, it
may not belong to
S
n
. Namely, some of
its elements may be beyond the borders
of the original constellation. Hence a
clipping-type operation is necessary and
this will introduce some loss.
SOFT DECISIONS
In practice, each symbol s
k
typically is
composed of information-carrying bits,
say
5
b
k, 1
, c, b
k, p
6
. It is then of interest
to take decisions on the individual bits
b
k,
i
, and often, also to quantify how reli-
able these decisions are. Such reliability
information about a bit is called a “soft
decision,” and is typically expressed via
the probability ratio
P
1
b
k,i
5 1| y
2
P
1
b
k,i
5 0| y
2
5
g
s:b
k,i
1
s
2
51
P
1
s| y
2
g
s:b
k,i
1
s
2
50
P
1
s| y
2
5
g
s:b
k,i
1
s
2
51
exp
a
2
1
s
||
y 2 Hs||
2
b
P
1
s
2
g
s:b
k,i
1
s
2
50
exp
a
2
1
s
||
y 2 Hs||
2
b
P
1
s
2
.
(9)
Here “
s
:b
k,i
1
s
2
5b” means all s for which
the ith bit of s
k
is equal to
b
, and P
1
s
2
is
the probability that the transmitter sent s.
To derive (9), use Bayes rule and the
Gaussian assumption made on e [4].
Fortunately, (9) can be relatively well
approximated by replacing the two sums
in (9) with their largest terms. To find
these maximum terms is a slightly
modified version of (2), at least if all s
are equally likely so that P
1
s
2
5 1/|S|
n
.
Hence, if (2) can be solved, good approx-
imations to (9) are available too. An
even better approximation to (9) is
obtained if more terms are retained, i.e.,
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